Reconfiguration of Inertia, Damping and Fault Ride-Through for a Virtual Synchronous Machine

ABSTRACT

This invention discloses a controller and method to reconfigure the inertia, damping and fault ride-through capability of a virtual synchronous machine (VSM). The virtual inertia is reconfigured via adding a low-pass filter to the torque (equivalently, active power) signal of the VSM and the virtual damping is reconfigured via adding a virtual damper between the VSM voltage and the output voltage sent to the PWM conversion, instead of adjusting the inertia and the damping in the swing equation. The function of the virtual damper is to scale the existing converter-side inductance or to insert the desired inductance into the VSM, which increases the damping. Moreover, the fault ride-through capability of the VSM can be reconfigured to achieve the given fault-current level via reconfiguring the damping and the inertia properly.

CROSS-REFERENCE TO RELATED APPLICATIONS

This nonprovisional patent application claims the benefit of and priority under 35 U.S. Code 119 (b) to U.K. Patent Application No. GB1800572.8 filed on Jan. 14, 2018, entitled “Reconfiguration of Inertia, Damping and Fault Ride-Through for a Virtual Synchronous Machine”, the contents of which are all hereby incorporated by reference herein in its entirety.

TECHNICAL FIELD

This invention is concerned with the control and operation of power electronic converters. Possible application fields include renewable energy, such as wind, solar and wave energy, electrical vehicles, energy storage systems, aircraft power systems, different types of loads that require power electronic converters, data centers etc.

BACKGROUND

Power systems are going through a paradigm change from centralized generation to distributed generation. More and more distributed energy resources (DERs), including renewables, electric vehicles, and energy storage systems, are being integrated into the grid. The integration of DERs presents unprecedented challenges and has become an important research topic in recent years. One challenge is that a DER unit often means low inertia or even inertia-less. The large-scale utilization of DERs would cause significant decrease of inertia, which brings considerable concerns about grid stability, because inertia has been regarded as a critical factor for guaranteeing the stability of power systems.

Since synchronous machines (SM) can provide large inertia because of the large kinetic energy stored in the rotors, a lot of efforts have been made in recent years to provide additional energy when needed to mimic the inertia. For example, a fast-response battery energy storage system can be adopted to inject additional power when needed. The inertia of a PV system can be increased by adjusting the DC-link voltage and the PV array output. The kinetic energy stored in the rotor of a wind turbine can be utilized for wind plants to participate in system frequency regulation.

Another important trend is to operate power electronic converters in DER units as virtual synchronous machines (VSM), which are power electronic converters that emulate the major features of a traditional SM, such as torque, inertia, voltage, frequency, phase, and field-excitation current. VSMs have become the building blocks for future power electronics-enabled autonomous power systems, which are characterized as synchronized and democratized (SYNDEM) smart grids. Different/similar options to implement VSMs have been proposed in the literature. The VISMA approach controls the inverter current to follow the current reference generated according to the mathematical model of SM, which makes inverters behave like controlled current sources. The synchronverter (SV) approach or the static synchronous generator disclosed in US 2011/0270463 A1 directly embeds the mathematical model of SM into the controller to control the voltage generated. The conventional inertia factor J and damping factor D_(p) of an SM are emulated through embedding the swing equation of SM in the controller. US 2014/0067138 A1 discloses a virtual controller of electromechanical characteristics for static power converters, which adopts a power loop controller with the capability of adjusting the inertia factor and the damping factor. The power loop controller is actually equivalent to the swing equation of SM. CN106208159A discloses a virtual synchronous machine-based dynamic power compensation method, which also adopts the swing equation of SM as the core of the controller but with the additional feature of adjusting the inertia factor and the damping factor according to the variation of the frequency. CN107154636A discloses a multi-target optimization control method, which also incorporates the swing equation of SM as the basis of the controller for optimization. In summary, a common feature of the state of the art about VSM is to incorporate the swing equation of SM and adjust the inertia factor and the damping factor accordingly. However, as to be shown later, the virtual inertia that can be provided by the swing equation of an SM is limited. Moreover, the frequency response of a VSM can be oscillatory when the virtual inertia increases. Furthermore, the output current of such a VSM can be excessive when a grid fault occurs, making it difficult to ride-through grid faults.

BRIEF SUMMARY

The following summary is provided to facilitate an understanding of some of the innovative features unique to the disclosed embodiments and is not intended to be a full description. A full appreciation of the various aspects of the embodiments disclosed herein can be gained by taking the entire specification, claims, drawings, and abstract as a whole.

After taking inventive steps, this invention discloses a controller and method to reconfigure the inertia, the damping and the fault ride-through capability of a VSM through a completely different way. Unlike the solutions found in the study of the state of the art, the disclosed controller and method does not adjust or optimize the inertia factor or the damping factor in the swing equation. Instead, an additional virtual inertia block and an additional virtual damping block are added to provide virtual inertia and virtual damping for a VSM.

In this disclosure, the SV or the static synchronous generator disclosed in US 2011/0270463 A1 is used as an example to facilitate the presentation. However, the disclosed invention can be applied to other schemes as long as there exists a channel that controls the active power (equivalently, the torque) and the frequency.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying figures further illustrate the disclosed embodiments and, together with the detailed description of the disclosed embodiments, serve to explain the principles of the present invention.

FIG. 1 shows a grid connected power electronic converter, i.e., the power part of a SV.

FIG. 2 shows the control part of a SV.

FIG. 3 shows the small-signal model of the active power loop of a conventional SV.

FIG. 4 shows the disclosed controller for a VSM to achieve reconfigurable inertia and damping through an additional virtual inertia block H_(v) and an additional virtual damper block.

FIG. 5 shows the small-signal model of the active-power loop with the reconfigurable virtual inertia block H_(v)(s) of the disclosed VSM.

FIG. 6 illustrates the implementations of a virtual damper: (a) through impedance scaling with a voltage controller; (b) through impedance insertion with a current controller.

FIG. 7 illustrates the normalized frequency response of a VSM with reconfigurable inertia and damping.

FIG. 8 shows the effect of the virtual damper on the frequency response.

FIG. 9 shows the fault ride-through capability of the VSM with different inertia: (a) the frequency response, and (b) the fault current.

DETAILED DESCRIPTION

The particular values and configurations discussed in these non-limiting examples can be varied and are cited merely to illustrate at least one embodiment and are not intended to limit the scope thereof.

The embodiments will now be described more fully hereinafter with reference to the accompanying drawings, in which illustrative embodiments of the invention are shown. The embodiments disclosed herein can be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art.

The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention. As used herein, the singular forms “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.

Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.

Subject matter will now be described more fully hereinafter with reference to the accompanying drawings, which form a part hereof, and which show, by way of illustration, specific example embodiments. Subject matter may, however, be embodied in a variety of different forms and, therefore, covered or claimed subject matter is intended to be construed as not being limited to any example embodiments set forth herein; example embodiments are provided merely to be illustrative. Likewise, a reasonably broad scope for claimed or covered subject matter is intended. Among other things, for example, subject matter may be embodied as methods, devices, components, or systems. Accordingly, embodiments may, for example, take the form of hardware, software, firmware or any combination thereof (other than software per se). The following detailed description is, therefore, not intended to be taken in a limiting sense.

Throughout the specification and claims, terms may have nuanced meanings suggested or implied in context beyond an explicitly stated meaning. used herein does not necessarily refer to a different embodiment. It is intended, for example, that claimed subject matter include combinations of example embodiments in whole or in part.

In general, terminology may be understood at least in part from usage in context. include a variety of meanings that may depend at least in part upon the context in which such terms are used. to mean A, B, and C, here used in the inclusive sense, as well as A, B, or C, here used in the exclusive sense. in part upon context, may be used to describe any feature, structure, or characteristic in a singular sense or may be used to describe combinations of features, structures or characteristics in a plural sense. to convey a singular usage or to convey a plural usage, depending at least in part upon context. to convey an exclusive set of factors and may, instead, allow for existence of additional factors not necessarily expressly described, again, depending at least in part on context.

Overview of the Synchronverter

In order to facilitate the presentation of the disclosure, the synchronverter (SV) is briefly introduced at first.

A synchronverter consists of two parts: the power part shown in FIG. 1 and the control part shown in FIG. 2. The power part is a conventional bridge converter cascaded with an LC or LCL filter. It can be of a single phase or of multiple phases. The converter-side inductor L_(s) and its parasitic resistance R_(s), then called converter-side impedance, mimic the stator inductor and its parasitic resistance of an SM. The control part includes the mathematical model of a synchronous generator described by

$\begin{matrix} {{{J\frac{d\; \omega}{d\; t}} = {T_{m} - T_{e} - {D_{p}\omega}}},} & (1) \\ {{T_{e} = {M_{f}i_{f}{\langle{i,{\sin\limits^{\sim}\; \theta}}\rangle}}},} & (2) \\ {{e = {\omega \; M_{f}i_{f}\sin\limits^{\sim}\; \theta}},} & (3) \\ {{Q = {{- {\omega M}_{f}}i_{f}{\langle{i,{\cos\limits^{\sim}\; \theta}}\rangle}}},} & (4) \end{matrix}$

where D_(p) represents the friction; T_(m), and T_(e) are the mechanical torque and the electromagnetic torque, respectively, θ is the rotor angle, w={dot over (θ)} is the angular frequency, M_(f)i_(f) reflects the field excitation, Q is the reactive power, and e is the generated VSM voltage or the virtual back electromotive force (EMF). Without loss of generality, it can be assumed that ω is also the frequency of the electricity generated under the assumption that the pair of poles per phase for the magnetic field is one. The vectors

θ and

θ are defined as

${{\sin\limits^{\sim}\; \theta} = \begin{bmatrix} {\sin \mspace{11mu} \theta} \\ {\sin \mspace{11mu} \left( {\theta - \frac{2\; \pi}{3}} \right)} \\ {\sin \mspace{11mu} \left( {\theta + \frac{2\; \pi}{3}} \right)} \end{bmatrix}},{{\cos\limits^{\sim}\; \theta} = \begin{bmatrix} {\cos \mspace{11mu} \theta} \\ {\cos \mspace{11mu} \left( {\theta - \frac{2\; \pi}{3}} \right)} \\ {\cos \mspace{11mu} \left( {\theta + \frac{2\; \pi}{3}} \right)} \end{bmatrix}}$

for three-phase applications or sin θ and cos θ for single-phase applications. The capacitor voltage v=[v_(a) v_(b) v_(c)]^(T) is the voltage of the SV, which is connected to the grid through a circuit breaker and a grid-side inductor. The equation (1) is called the swing equation, where J is called the inertia and D_(p) plays the role of the damping. The three equations (2)-(4) are included in the Calculation block of FIG. 2. Note that the torque T_(e) and the active power P of a synchronous machine satisfy P=ωT_(e). When the frequency ω only varies within a small range, what holds true for the torque T_(e) equivalently holds true for the active power P.

The controller shown in FIG. 2 has two channels: the frequency channel to generate the frequency and the voltage channel to generate the field-excitation M_(f)i_(f), which together form the voltage amplitude E of e. The term D_(p) actually serves the purpose of the frequency droop control and the term D_(q) is introduced to implement the voltage droop control. The integrator

$\frac{1}{K_{s}}$

generates the field exicitation M_(f)i_(f). In the steady state, there are

${\omega = {\omega_{ref} + {\frac{1}{D_{p}}\left( {T_{m} - T_{e}} \right)}}},{V_{m} = {V_{ref} + {\frac{1}{D_{q}}{\left( {Q_{set} - Q} \right).}}}}$

As a result, an SV is able to take part in the regulation of system frequency and voltage. Since the commonly-needed phase-locked loop can be removed, it is not shown in the controller.

The frequency droop coefficient D_(p) is defined for the frequency drop of α (%) to cause the torque (equivalently, active power) increase of 100%. Then,

$\begin{matrix} {{D_{p} = {\frac{S_{n}/\omega_{n}}{{\alpha\omega}_{n}} = \frac{S_{n}}{{\alpha\omega}_{n}^{2}}}},} & (5) \end{matrix}$

where ω_(n) is the rated angular frequency and S_(n) is the rated power. The voltage droop coefficient D_(q) is defined for the voltage drop of β (%) to cause the reactive power increase of 100%. Then,

$\begin{matrix} {{D_{q} = \frac{S_{n}}{\beta \sqrt{2}V_{n}}},} & (6) \end{matrix}$

where V_(n), is the rated RMS phase voltage. The √{square root over (2)} is due to the fact that the voltage amplitude V_(m) instead of the RMS voltage is fed back. v_(f) is the feedback voltage, which is often v.

Limitation of the Inertia of a Synchronverter

Under some mild conditions, the two channels can be decoupled by design. The small-signal model of the frequency channel can be illustrated as shown in FIG. 3. The gain K_(pd) reflects the amplification from the change of the power angle, Δδ, to the change of the active power and can be represented as

${K_{p\; d} = {N\frac{V_{n}^{2}}{X}}},$

where X=ω_(n)L_(s) is the impedance of the inductor L_(s), and N represents the number of phases, i.e., N=3 for a three-phase VSM or N=1 for a single-phase VSM. The transfer function from ΔP to Δω is then

$\begin{matrix} {{\frac{\Delta \; \omega}{\Delta \; P} = {{- \frac{1}{\omega_{n}D_{p}}} \cdot \frac{1}{{\tau_{\omega}s} + 1}}},} & (7) \end{matrix}$

where τ_(ω)=J/D_(p) is called the inertia time constant as mentioned before. Naturally, J=τ_(ω)D_(p) is often regarded as the virtual inertia. However, it also determines the time constant of the frequency loop. Hence, the inertia provided by J may be limited, as explained below.

The transfer function from ΔP_(set) to ΔP can be found as

$\begin{matrix} {{\frac{\Delta \; P}{\Delta \; P_{set}} = \frac{1}{{{\tau_{p}\left( {{\tau_{\omega}s} + 1} \right)}s} + 1}},} & (8) \end{matrix}$

where

$\begin{matrix} {\tau_{p} = {\frac{D_{p}\omega_{n}}{K_{p\; d}} = {\frac{D_{p}\omega_{n}X}{N\; V_{n}^{2}}.}}} & (9) \end{matrix}$

Substituting (5) into it, then

$\begin{matrix} {\tau_{p} = {\frac{S_{n}X}{\alpha \; N\; V_{n}^{2}\omega_{n}}.}} & (10) \end{matrix}$

Assume the base voltage is chosen as the RMS phase voltage V_(n) and the base power is chosen as the rated power S_(n). Then the base impedance is

Z _(base) =NV _(n) ² /S _(n)

and the per-unit output impedance of the SV is

$X^{pu} = {\frac{X}{Z_{base}} = {\frac{S_{n}X}{N\; V_{n}^{2}}.}}$

Hence, there is

$\begin{matrix} {\tau_{p} = {\frac{X^{pu}}{\alpha \; \omega_{n}}.}} & (11) \end{matrix}$

Since ω_(n) and α are both specified by the grid code, it is obvious that, for a given power system, τ_(p) is only determined by and proportional to X^(pu). Once the SV hardware is designed, the corresponding τ_(p) is fixed.

The system in FIG. 3 or the transfer function (8) has two eigenvalues

$\lambda_{1,2} = {\frac{{- \tau_{p}} \pm \sqrt{\tau_{p}^{2} - {4\; \tau_{p}\tau_{\omega}}}}{2\tau_{p}\tau_{\omega}} = {\frac{{- 1} \pm \sqrt{1 - {4\; {\tau_{\omega}/\tau_{p}}}}}{2\; \tau_{\omega}}.}}$

When 0<τ_(ω)<<T_(p), the two eigenvalues are on the real axis, at

$\lambda_{1} \approx {{- \frac{1}{\tau_{\omega}}}\mspace{14mu} {and}\mspace{14mu} \lambda_{2}} \approx {- {\frac{1}{\tau_{p}}.}}$

The system is dominated by λ₂ because λ₁ is too far away from the imaginary axis. When τ_(ω) increases, the eigenvalues move towards each other on the real axis. Then the two eigenvalues become conjugate when τ_(ω)>0.257τ_(p), with the imaginary part initially increasing and then decreasing. The imaginary part reaches the maximum when τ_(ω), =0.57τ_(p). The two eigenvalues move toward the imaginary axis when τ_(ω) increases further. The damping ratio of the active-power loop when τ_(ω)>0.25τ_(p) is

$\begin{matrix} {Ϛ = {\frac{1}{2\sqrt{\tau_{\omega}/\tau_{p}}}.}} & (12) \end{matrix}$

Note that

$Ϛ = \frac{1}{\sqrt{2}}$

when τ_(ω)=0.5τ_(p). In other words, increasing τ_(ω) makes the system response oscillatory and reduces the stability margin. This means there is an upper limit on the inertia time constant τ_(ω) for a given system, which has a fixed τ_(p) as given in (11). In other words, the virtual inertia provided by J or τ_(ω) is limited.

Indeed, since γ_(ω) is the time constant of the frequency loop, it is recommended to chose it much smaller than the fundamental period, e.g., as 0.1/f_(n) s. As a result, the condition 0<τ_(ω)<<τ_(p) often holds. The transfer function from ΔP_(set) to ΔP given in (8) can be simplified as

$\begin{matrix} {\frac{\Delta \; P}{\Delta \; P_{set}} = {\frac{1}{{\tau_{p}s} + 1}.}} & (13) \end{matrix}$

The Disclosed Invention

FIG. 4 shows the disclosed controller and method for a VSM to achieve reconfigurable inertia and damping. A virtual inertia block Hu (s) is added to the torque (equivalently, active power) signal of the VSM and a virtual damper is added to the VSM voltage e before sending it out as output voltage e_(PWM). As will be shown later, these two are linked together. The voltage feedback can be chosen as v, e, or e_(PWM), depending on the implementation. The integrators can be implemented with the normal integrator, or the normal integrator with saturation, or a nonlinear integrator that is able to limit the output of the integrator.

The Virtual Inertia

In order to be able to reconfigure the inertia of a VSM, a virtual inertia block H_(v)(s) is added to the torque signal of the VSM, as shown in FIG. 4, instead of adjusting the inertia J in the swing equation. It can be implemented via a low-pass filter to slow the response down. There are many options to implement this and the simplest one is to adopt

$\begin{matrix} {{{H_{v}(s)} = \frac{1}{{J_{v}s} + 1}},} & (14) \end{matrix}$

where J_(v) is the virtual inertia required. There is normally a low-pass filter to remove the ripples in the torque but its time constant is often much smaller than J_(v) and it has a different function. The virtual block H_(v) can be put in series with it or replace it.

The corresponding small-signal model of the active-power loop is shown in FIG. 5. The transfer function from ΔP to Δω is

$\frac{\Delta\omega}{\Delta \; P} = {{- \frac{1}{\omega_{n}D_{p}}} \cdot \frac{1}{{\tau_{\omega}s} + 1} \cdot {\frac{1}{{J_{\upsilon}s} + 1}.}}$

This is a second-order system with a small τ_(ω) as explained before. It can be simplified as

$\begin{matrix} {\frac{\Delta\omega}{\Delta \; P} \approx {{- \frac{1}{\omega_{n}D_{p}}} \cdot {\frac{1}{{\left( {J_{\upsilon} + \tau_{\omega}} \right)s} + 1}.}}} & (15) \end{matrix}$

If J_(v)>>T_(ω), then the equivalent inertia is J_(v).

The characteristic equation of the active-power loop shown in FIG. 5 can be found as

1+τ_(p)(J _(v) s+1)(τ_(ω) s+1)s=0  (16)

If a relatively large inertia J_(v)>>τ_(ω) is desired, then the term

$\frac{1}{{\tau_{\omega}s} + 1}$

can be ignore and the characteristic equation (16) can be simplified as

τ_(p)(J _(v) s+1)s+1=0.

This is the same as the characteristic equation of the system in (8) but with τ_(ω) replaced by J_(v). According to (12), the damping ratio of the system when J_(v)>0.25τ_(p) is

$\begin{matrix} {\zeta = {\frac{1}{2\sqrt{J_{\upsilon}/\tau_{p}}}.}} & (17) \end{matrix}$

If the virtual inertia J_(v) is configured to be large with comparison to τ_(p), then the damping of the system is small, resulting in oscillatory responses. This may lead to large transient currents, which might overload or even damage the converter. It is critical for the damping of the VSM to be large enough.

Assume the desired damping ratio is ζ₀. Then, according to (17), the corresponding τ_(p) is

τ_(p)=4ζ₀ ² J _(v)

and, according to (11), the equivalent p.u. impedance X_(v) ^(pu) is

X _(v) ^(pu)=αω_(n)τ_(p)=4ζ₀ ² J _(v)αω_(n).  (18)

This requires the corresponding inductance L_(v) to be

$\begin{matrix} {L_{\upsilon} = {{\frac{X_{\upsilon}^{pu}}{\omega_{n}}Z_{base}} = {{4\zeta_{0}^{2}J_{\upsilon}\alpha \; Z_{base}} = {\frac{4\zeta_{0}^{2}J_{\upsilon}\alpha \; {NV}_{n}^{2}}{S_{n}}.}}}} & (19) \end{matrix}$

Apparently, this is different from the inductance L_(s) existing in the hardware.

The Virtual Damper

As discussed, there is a need to reconfigure the damping of the VSM to avoid oscillatory frequency responses, which boils down to reconfigure the converter-side inductance L_(s) as L_(v). This can be achieved via putting the generated VSM voltage e through a virtual damper before sending it for PWM conversion, as shown in FIG. 4, instead of adjusting the damping D_(p) in the swing equation. The virtual damper also takes the voltage v and/or the current i as inputs. Here, two possible implementations are shown in FIG. 6, one through impedance scaling with a voltage feedback controller and the other through impedance insertion with a current feedback controller.

Through Impedance Scaling with an Inner-loop Voltage Controller

As shown in FIG. 6(a), it consists of a voltage feedback controller to scale the voltage difference e−v with the factor D and subtracts it from the original signal e to form the new control signal e_(PWM). Hence,

e _(PWM) =e−D(e−v),  (20)

which, in lieu of e, is converted to PWM signals to drive the switches in the power part. Since the switching frequency of the converter is normally much higher than the system frequency, there is

e _(PWM) ∞v+v ₈,  (21)

where v_(s) is the voltage across the inductor L_(s), when considering the average values over a switching period for the PWM signals. Combining (20) and (21), then there is

$\begin{matrix} {e \approx {\upsilon + {\frac{1}{1 - D}{\upsilon_{s}.}}}} & (22) \end{matrix}$

In other words, the function of the virtual damper is to replace the inductor L_(s) with

$\frac{1}{1 - D}L_{s}$

or to scale the original impedance by

$\frac{1}{1 - D}.$

Hence, this technique is called impedance scaling. In order to scale L_(s) to L_(v), D should be chosen as

$D = {1 - \frac{L_{s}}{L_{\upsilon}}}$

or, substituting (19) into it, as

$D = {1 - {\frac{S_{n}L_{s}}{4\zeta_{0}^{2}J_{\upsilon}\alpha \; {NV}_{n}^{2}}.}}$

The gain

$D = {1 - \frac{L_{s}}{L_{\upsilon}}}$

is static but D can also be chosen dynamic as well because the function of the virtual damper is to scale the original impedance by

$\frac{1}{1 - D},$

which can be designed to include desired frequency characteristics. For example, it can be chosen as

${D(s)} = {2 - \frac{L_{s}}{L_{\upsilon}} - {\Pi_{h}\frac{s^{2} + {2\zeta_{h}h\; \omega_{n}K_{h}s} + \left( {h\; \omega_{n}} \right)^{2}}{s^{2} + {2\zeta_{h}h\; \omega_{n}s} + \left( {h\; \omega_{n}} \right)^{2}}}}$

where ζ_(h) can be chosen as ζ_(h)=0.01 to accommodate frequency variations and h can be chosen to cover the major harmonic components in the current, e.g. the 3rd, 5th and 7th harmonics. The scaling factor is

$\frac{1}{1 - D} = {\frac{1}{1 - \left( {2 - \frac{L_{s}}{L_{\upsilon}} - 1} \right)} = \frac{L_{\upsilon}}{L_{s}}}$

at low and high frequencies and

$\frac{1}{1 - D} = {\frac{1}{1 - \left( {2 - \frac{L_{s}}{L_{\upsilon}} - K_{h}} \right)} = \frac{1}{K_{h} - 1 + \frac{L_{s}}{L_{\upsilon}}}}$

at the h-th harmonic frequency. While it meets the requirement of the virtual damping, it also scales the impedance at the harmonic frequencies by a factor of

$\frac{1}{K_{h} - 1 + \frac{L_{s}}{L_{v}}}.$

If

${K_{h} > {2 - \frac{L_{s}}{L_{v}}}},$

then the impedance at the h-th harmonic frequency is reduced, which enhances the quality of the VSM voltage v.

Through Impedance Insertion with an Inner-Loop Current Controller

Instead of using the voltage feedback controller shown in FIG. 6(a), it is also possible to adopt a current feedback controller to implement the virtual damper and generates the new control signal e_(PWM), as shown in FIG. 6(b). The voltage difference e−e_(PWM) is passed through an impedance Z(s) to generate a current reference, of which the difference with the feedback current i is scaled by a factor F and added to the original signal e to form the new control signal e_(PWM). Hence,

$\begin{matrix} {{e_{P\; W\; M} = {e + {F\left( {{\frac{1}{Z(s)}\left( {e - e_{P\; W\; M}} \right)} - i} \right)}}},} & (23) \end{matrix}$

which, in lieu of e, is converted to PWM signals to drive the switches in the power part. This is equivalent to having

$e = {e_{P\; W\; M} + {\frac{F}{{Z(s)} + F}{Z(s)}{i.}}}$

Choose F as a positive large number and

Z(s)=sL _(v).  (24)

Then

$\frac{F}{{s\; L_{v}} + F}$

is a low-pass filter with a small time constant and

$\frac{F}{{s\; L_{v}} + F} \approx 1$

over a wide range of frequencies. As a result,

$e = {{e_{P\; W\; M} + {\frac{F}{{s\; L_{v}} + F}s\; L_{v}i}} \approx {e_{P\; W\; M} + {s\; L_{v}{i.}}}}$

In other words, the function of the virtual damper shown in FIG. 6(b) is to insert an inductor L_(v) between e and e_(PWM) with the current i flowing through it, meeting the requirement on the equivalent inductance. Hence, this technique is called impedance insertion. Here, it is assumed that e_(PWM) is adopted as the feedback voltage v_(f). If v_(f)=v, then the inductance L_(s) between e_(PWM) and v should be considered, via choosing Z(s)=s(L_(v)−L_(s)).

The impedance Z(s) in (24) is inductive but it can include a resistive term as well.

Fault Ride-Through Capability

The fault ride-through capability of a VSM is very important. The worst case is that there is a ground fault across the capacitor, i.e., v=0. In this case, the whole voltage e is dropped on the corresponding equivalent inductance L_(v). Since the voltage v is dropped to nearly zero, the corresponding reactive power in the steady state (assuming Q_(set)=0) is

$Q = {{D_{q}V_{ref}} = {{\frac{S_{n}}{\beta \sqrt{2}V_{n}}\sqrt{2}V_{n}} = \frac{S_{n}}{\beta}}}$

when the parasitic resistance R_(s) is negligible. This is the reactive power of the equivalent inductance L_(v). Hence, the corresponding p.u. RMS voltage E_(fault) ^(pu) is

$E_{fault}^{pu} = {{\sqrt{\frac{{Q\omega L}_{v}}{N}}/V_{n}} = \sqrt{\frac{S_{n}{\omega L}_{v}}{\beta \; {NV}_{n}^{2}}.}}$

Note that this voltage is a number and it is not a physical voltage so even if it is large it is not a problem. Substituting (19) into it and considering that ω≈ω_(n) and (18), then

${E_{fault}^{pu} \approx \sqrt{\frac{S_{n}\omega}{\beta \; {NV}_{n}^{2}}\frac{4\; Ϛ_{0}^{2}J_{v}\alpha \; N\; V_{n}^{2}}{S_{n}}}} = {\sqrt{\frac{4\; Ϛ_{0}^{2}J_{v}{\alpha\omega}_{n}}{\beta}} = {\sqrt{\frac{X_{v}^{pu}}{\beta}}.}}$

As a result, the corresponding p.u. RMS fault current is

$\begin{matrix} {I_{fault}^{pu} = {\frac{E_{fault}^{pu}}{X_{v}^{pu}} = {\frac{1}{\sqrt{\beta \; X_{v}^{pu}}}.}}} & (25) \end{matrix}$

In practice, the actual E_(fault) ^(pu) would be larger and the actual I_(fault) ^(pu) would be smaller because of the parasitic resistance R_(s). The larger the voltage droop coefficient β, the smaller the fault current; the larger the equivalent p.u. impedance X_(v) ^(pu) (or the inertia) the smaller the fault current.

TABLE I Simulation parameters Parameters Values Parameters Values τ_(ω) 0.002 s L_(s) 0.23 mH J_(ν) 0.02 s P_(set) 50 W D_(p) 0.2026 Q_(set) 0 Var D_(q) 117.88 Nominal power 100 VA V_(n) 12 V f_(n) 50 Hz

For a desired fault current I_(fault) ^(pu), according to (25), the required X_(v) ^(pu) is

$X_{v}^{pu} = \frac{1}{{\beta \left( I_{fault}^{pu} \right)}^{2}}$

and the corresponding L_(v) is

$L_{v} = {\frac{X_{v}^{pu}Z_{base}}{\omega_{n}} = {\frac{1}{{\beta \left( I_{fault}^{pu} \right)}^{2}}{\frac{N\; V_{n}^{2}}{\omega_{n}S_{n}}.}}}$

According to (18), the corresponding inertia J_(v) is

$J_{v} = {\frac{X_{v}^{pu}}{4\; Ϛ_{0}^{2}\alpha \; \omega_{n}} = {\frac{1}{4\; Ϛ_{0}^{2}\alpha \; \omega_{n}{\beta \left( I_{fault}^{pu} \right)}^{2}}.}}$

This is a very fundamental formula. It links together most of the key parameters of the disclosed VSM, including the damping ratio ζ₀, the virtual inertia J_(v), the frequency droop coefficient α, the voltage droop coefficient β, the fault current level I_(fault) ^(pu), and the rated system frequency ω_(n). These are all linked with each other.

In implementation, it is possible to limit the integrator output M_(f)i_(f) to limit E_(fault) ^(pu) and, hence, the fault-current level I_(fault) ^(pu). If e is adopted as the voltage feedback v_(f), it is also possible to reduce the fault-current level I_(fault) ^(pu).

Validation with Computational Simulations

The parameters of the single-phase converter used in the simulations are given in Table I. The frequency droop coefficient is chosen as α=0.5%, which leads to D_(p)=0.2026, and the voltage droop coefficient is chosen as β=5%, which leads to D_(q)=117.88. The desired damping ratio is chosen as ζ₀=0.707. As a result,

$L_{v} = {\frac{{2J_{v}\alpha \; N\; V_{n}^{2}}\;}{S_{n}} = {{\frac{2J_{v} \times 0.5\% \times 12^{2}}{100} \times 1000} = {14.4\; J_{v}\mspace{14mu} {{mH}.}}}}$

Reconfigurability of the Inertia and the Damping

In this case, the VSM operates in the stand-alone mode to supply both LD₁ and LD₂ equal to 0.4 pu via the AC bus. Before t=0, both LD₁ and LD₂ are connected to the AC bus so the load power is 0.8 pu. At t=0, LD₂ is disconnected from the AC bus. The damping is configured as ζ₀=0.707.

The frequency responses of the VSM with different virtual inertia are shown in FIG. 7. When the load LD2 is disconnected, the frequency increases. Apparently, the frequency response behaves as expected: increasing the virtual inertia J_(v) indeed slows the frequency response down. Moreover, because of the damping ratio is designed to be ζ₀=0.707, the frequency response is very smooth, without visible overshoot. Note that the steady-state frequencies before and after the load change in the two cases are slightly different because the actual real power are slightly different.

Effect of the Virtual Damping

In this case, the VSM is connected to a stiff grid to supply both LD₁ and LD₂ equal to 0.4 pu. Before t=0, the active power set point of the VSM is 0. At t=0, it is changed to ΔP_(set)=50 W. The simulation results with virtual inertia J_(v)=0.5 s are shown in FIG. 8 for the cases with and without the virtual damper. When the virtual damper is not enabled, the response is oscillatory (denoted “Without” in the figure). When the virtual damper is enabled, the response is very smooth (denoted “With damper” in the figure).

Fault Ride-Through Capability

In this case, the VSM is connected to supply both LD₁ and LD₂ equal to 0.4 pu. At t=0, a ground fault occurs at the AC bus and lasts for 0.5s. The frequency responses with different inertia J_(v) under the ground fault are shown in FIG. 9. Before the ground fault occurs, the frequencies at the two cases are slightly different because of the slight difference of the active power output. In both cases, the frequency changes slightly but the current changes dramatically and over-current appears. The larger the inertia, the smaller the over-current. With J_(v)=1 s, the current is about 236%, which is slightly less than the value of 252% calculated from (25). After the fault is cleared at t=0.5 s, the frequency and the current of the VSMs return to normal. The provision of reconfigurable inertia J_(v) and damping can significantly reduce the over-current, and enhance the fault ride-through capability.

It will be appreciated that variations of the above-disclosed and other features and functions, or alternatives thereof, may be desirably combined into many other different systems or applications. It will also be appreciated that various presently unforeseen or unanticipated alternatives, modifications, variations or improvements therein may be subsequently made by those skilled in the art, which are also intended to be encompassed by the following claims. 

What is claimed is: 1) A controller and method to reconfigure the inertia, damping and fault ride-through capability of a power electronic converter having an LC filter consisting of a converter-side inductor and a capacitor that is operated as a virtual synchronous machine (VSM) to emulate the features of a conventional synchronous machine such as torque, inertia, voltage, frequency, phase, and field-excitation current, comprising a virtual inertia block that takes the torque (equivalently, the active power) signal as the input and generates a torque (equivalently, active power) feedback signal; and a virtual damper that takes the VSM voltage, a feedback voltage and/or the converter-side inductor current from the power electronic converter as inputs to generate an output voltage to drive the power electronic converter after PWM conversion. 2) A controller and method as claimed in claim 1 further comprises a frequency channel that takes the difference between a given torque (equivalently, active power) reference and the torque (equivalently, active power) feedback signal to generate a frequency signal and, furthermore, a phase signal after integrating the frequency signal; a voltage channel that takes the difference between a given reactive power reference and a reactive power feedback signal, with the option of adding the scaled difference between a voltage reference and a voltage feedback signal, to generate a field-excitation signal; and a calculation block that takes the frequency signal and the phase signal generated in the frequency channel, the field-excitation signal generated in the voltage channel, and the current flowing through the converter-side inductor as inputs to generate, according to the mathematical model of a synchronous machine, the VSM voltage, the torque (equivalently, active power) signal and the reactive power feedback signal for the voltage channel. 3) A controller and method as claimed in claim 1 in which the virtual inertia block is a low-pass filter. 4) A controller and method as claimed in claims 1 and 3 in which the virtual inertia block is a first-order low-pass filter with its time constant reconfigured as a given inertia time constant. 5) A controller and method as claimed in claim 1 in which the virtual damper reconfigures the converter-side inductor to meet the requirement of the given inertia time constant and a given damping. 6) A controller and method as claimed in claims 1 and 5 in which the virtual damper functions as scaling the converter-side inductor. 7) A controller and method as claimed in claims 1, 5 and 6 in which the virtual damper generates the output voltage via subtracting from the VSM voltage the difference between the VSM voltage and the capacitor voltage scaled by a factor. 8) A controller and method as claimed in claim 7 in which the factor is a static gain that is determined in such a way to achieve the given inertia and the given damping. 9) A controller and method as claimed in claim 7 in which the factor is a dynamic transfer function. 10) A controller and method as claimed in claims 1 and 5 in which the virtual damper functions as inserting a desired impedance between the VSM voltage and the output voltage to meet the requirement of the given inertia and the given damping. 11) A controller and method as claimed in claims 1, 5 and 10 in which the virtual damper generates the output voltage via adding, to the VSM voltage, the scaled difference between the reference current generated by passing the voltage difference between the VSM voltage and the output voltage through an impedance and the converter-side inductor current. 12) A controller and method as claimed in claim 11 in which the impedance is inductive with the value determined according to the inductance that meets the requirement of the given inertia and the given damping. 13) A controller and method as claimed in claim 11 in which the impedance is inductive and resistive. 